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Solid Oxide Fuel Cell (SOFC) Assessment
Our fuel cell assessment model is included here.
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Index of assessment methods |
Fuel cell assessment method stages
The following calculation stages are incorporated in the spreadsheets.
Reforming reaction
Undertake calculations for a unit mass flow rate of hydrogen from the reforming reaction to the fuel cell reaction.
Select fuel and write the reforming reaction equations including the water gas shift reaction.
Undertake the molar and mass balance for the reforming reaction and calculate the theoretical mass ratio of fuel in, to hydrogen out. Increase the fuel in by the reformer efficiency (difference due to equilibrium position and fuel used for preheating):
| eR = | Theoretical fuel flow required to yield unit flow of hydrogen |
| Actual fuel flow in to yield unit flow of hydrogen |
Calculate the heat required by the reforming process from the enthalpy of reaction:
DHref = S [DHformation-products - DHformation-reactants]
Gibbs energy and enthalpy of fuel cell reaction
Write the fuel cell reaction equation.
Undertake the molar and mass balance for the fuel cell reaction.
Calculate the max theoretical electrical power from the Gibbs energy at operating T & P [ref 1]:
Wout = -DG = -DGt + R.T.[ ln (product of partial pressures of products)
Where DGt = R.T. ln Kp, with ln Kp taken from tables for the operating temp [ref 2].
Partial pressures are relative to 1 bar datum, and the anode and cathode streams are assumed separate due to the solid electrolyte. The partial pressure of the oxygen is assumed equal to the partial pressure of the air.
Open circuit cell voltage:
Voc = -DG/(ne.F)
Heat of fuel cell reaction:
DHfc = S [DHformation-products - DHformation-reactants]
Fuel cell characteristics and efficiency factors
The voltage - current characteristic of fuel cells derives from:
- Voltage losses due to internal resistance and electrode activation potentials represented by the "voltage efficiency"
- Current losses due the effect of fluid flows and concentrations on the utilisation ratio of the fuel represented by the "utilisation resistance" or "current efficiency"
Internal resistance losses dominate except at high current densities or high fuel utilisation ratios (>80%), when mass transfer effects, or low concentrations / partial pressures of fuel cause a fall off of voltage. Other concentration effects include humidification of the anode stream and use of air in place of oxygen.
These effects are dependent upon the detailed construction of the fuel cell and the fluid flow channels, and are hence difficult to model theoretically. However, due to the dominance of internal resistance the characteristic is substantially linear over normal operating conditions and may be derived empirically.
Given suppliers recommended operating characteristics of the fuel cell, or using characteristics given in the model derived from results on development fuel cells given in research papers [refs 3,4], specify the target operating current density and operating voltage to give a good combination of power output, voltage efficiency, and fuel utilisation.
The fuel cell voltage - current characteristic may be approximated to a linear relation [ref 5], determined by the target operating point and open circuit voltage from y=mx+c, or may be more accurately represented by an empirical relationship [refs 6,7] of the form:
V = Voc - R.i - b. ln (i) - m. e ni
The characteristics are plotted to aid specification of a suitable operating point.
The voltage efficiency is calculated from ev = V/Voc and the current efficiency from:
ei = (current density x cell area) /(mass flow of hydrogen x ne x F)
The theoretical efficiency is calculated from g = DG/DH for the fuel cell reaction, and the electrical generation efficiency calculated:
ee = g ev ei eR eC
Where eC is the DC/AC converter efficiency.
It is assumed that a constant DC voltage is required for conversion to a constant AC voltage, and to keep the power conditioner simple, that excess DC voltage is dropped through a resistor. Thus the DC conversion voltage is equal to the fuel cell voltage at the rated or maximum required operating current, and the voltage efficiency remains constant partial loads.
Due to the rapid drop off of voltage at low fuel concentrations, regulation of fuel flow by voltage could be unstable and regulation of fuel flow proportional to current is assumed. Thus the fuel utilisation ratio or current efficiency is also constant.
Ancillary loads such as external preheaters, pumps, fans, and compressors are identified.
Electricity and heat output v fuel consumption
Two options are available:
- Calculating the heat output and fuel consumed for a given electrical power output
- Calculating the electricity and heat output for a given supply of fuel
| Fuel consumed for a required electrical output | Electricity output for an available fuel supply | |||||||||||||||
| Add ancillary loads referred to AC supply, & add converter efficiency factor to give DC output required from fuel cell stack.
Determine sum of currents through all cells in stack:
Hence hydrogen utilised in stack:
Hence hydrogen flow from utilisation efficiency. Fuel flow to reformer from fuel to hydrogen mass ratio, and reformer efficiency. Total fuel consumption including external preheating. |
Total fuel available less fuel used in external preheating. Hydrogen flow from reformer by reformer efficiency, and hydrogen to fuel mass ratio. Hence hydrogen utilised in fuel cell stack from utilisation efficiency. Determine sum of currents through all cells in stack: SCell currents = Mass flow x charge per unit mass
Stack power = Cell voltage x SCell currents Hence DC output and AC output from converter efficiency. Total AC power out less ancillary loads referred to AC output. |
Heat output
From SFEE
Qin + Win = Qout + Wout + DH
Or Qout - Qin = DG - DHfc - DHref (DG and DHfc are negative)
Qin includes reformer preheating and preheating external to the reformer or fuel cell stack.
Assuming a proportion "Rec" of voltage losses within the fuel cell stack are recovered as heat, and correcting for fuel utilisation ei
Qout - Qin = [ (DG - DHfc). ei ] + [ - Rec.(1- ev). DG. ei ] - DHref
When unutilised fuel is recirculated and burnt to preheat the fuel internally:
Qout - Qin = [ (DG - DHfc). ei ] + [ -Rec.(1- ev). DG. ei ] - { DHref - [ ( 1 - ei ). DHcombustion ]}
Where DHcombustion = DHformation H2O (vap) at the high temperatures being considered
The total heat output and the heat gain are reduced by heat losses from the external preheater, fuel cell stack, and from losses up to the heat exchanger. The usable heat output and heat gain are then further reduced by the effectiveness of the heat exchanger.
The heat efficiency is calculated as:
| et = | Net usable heat gain | = | (Qout - Qin external - losses) x heat exchanger effectiveness |
| Heat value of fuel in | DHfc/reformer efficiency |
The denominator is consistent with the electrical generation efficiency and the combined heat and power efficiency is calculated:
ehp = ee + et
The heat to power ratio is calculated from:
| Rhp = | Total heat out | = | Qout - losses |
| Total electricity out | Total electricity output |
Fuel stack configuration
Two calculation options are available:
- Specify number of cells in series to determine fuel cell stack voltage and number of cells required in parallel to supply power
- Specify desired fuel cell stack operating voltage to determine numbers of cells required in series and parallel
When the desired operating voltage gives a non-integer value of cells in series, an integer value is chosen and the stack voltage modified. The cell voltage is kept to that originally specified in this case as a small difference in cell operating voltage could cause a large and undesirable change in cell operating current.
When the number of cells in parallel to meet the required power is non-integer, an integer value is chosen and the cell operating current adjusted to suit. The resulting operating point may then be compared with the voltage-current characteristic.
Input of the cell thickness, the cell area, and the total number of cells enables the core volume of the fuel cell stack to be calculated, although this will only be applicable to the rectangular sandwich type constructions.
Air flow required for fuel cell reaction, stack cooling and heat transfer
Water flows
The balance of water emitted from the fuel cell reaction, and that consumed by the reforming reaction is calculated from the mass balances of the reaction equations and the fuel utilisation factor. This neglects the flow of water required to the fuel cell anode required to prevent damage of the anode due to dehydration.
References
- S Zumdahl; Chemistry; Houghton Mifflin 1997
- G Rogers & Y Mayhew; Thermodynamic and Transport Properties of Fluids; Blackwell
- P Costamagna & K Honegger; Modeling of Solid Oxide Heat Exchanger Integrated Stacks and Simulation at High Fuel Utilization; Journal of the Electrochemical Society, Vol 145, No 11, 1998
- F Bevc; Advances in solid oxide fuel cells and integrated power plants; Journal of power and energy, Part A, Proceedings, Institution of Mechanical Engineers, Vol 211, 1997
- F Standaerdt, K Hemmes, N Woudstra; Analyticl fuel cell modeling; Journal of Power Sources 63, 1996
- T Swaddle; Inorganic Chemistry; Academic Press 1997
- J Lee & T Lalk; Modeling fuel cell stack systems; Journal of Power Sources 73, 1998
- F Gardner; Thermodynamic processes in solid oxide and other fuel cells; Journal of power and energy, Part A, Proceedings, Institution of Mechanical Engineers, Vol 211, 1997
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